Problem 18 - College Physics

Question

\(\bullet\) Threshold of vision. Under controlled darkened conditions in the laboratory, a light receptor cell on the retina of a person's eye can detect a single photon (more on photons in Chapter 28 ) of light of wavelength 505 \(\mathrm{nm}\) and having an energy of \(3.94 \times 10^{-19} \mathrm{J} .\) We shall assume that this energy is absorbed by a single cell during one period of the wave. Cells of this kind are called rods and have a diameter of approximately 0.0020 \(\mathrm{mm} .\) What is the intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) delivered to a rod?

Step-by-Step Solution

Answer

The intensity is approximately \(1.25 \times 10^{-7} \text{ W/m}^2\).

Step 1: Convert the diameter to meters

The diameter of the rod is given as 0.0020 mm. To use it in our calculations with standard SI units, we need the diameter in meters. We convert millimeters to meters by dividing by 1000: \(0.0020 \text{ mm} = 0.0020 \times 10^{-3} \text{ m} = 2.0 \times 10^{-6} \text{ m}\).

Step 2: Calculate the area of the rod

The rod is assumed to be circular in cross-section. We calculate the area \(A\) using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius. The radius is half the diameter: \(r = \frac{2.0 \times 10^{-6} \text{ m}}{2} = 1.0 \times 10^{-6} \text{ m}\). Substituting this into the formula gives \(A = \pi \times (1.0 \times 10^{-6} \text{ m})^2 = \pi \times 1.0 \times 10^{-12} \text{ m}^2\).

Step 3: Calculate the intensity

Intensity \(I\) is defined as power per unit area. Here the power \(P\) absorbed by the rod is equal to the energy of the photon, \(3.94 \times 10^{-19} \text{ J}\), over the duration of one wave period. Assuming one photon is absorbed, we take the energy value directly as the power. Thus, \(I = \frac{3.94 \times 10^{-19} \text{ W}}{\pi \times 1.0 \times 10^{-12} \text{ m}^2}\). Calculating this gives \(I \approx 1.25 \times 10^{-7} \text{ W/m}^2\).

Key Concepts

Intensity Calculation

Intensity is a measure of how much power is delivered over a specific area. In the context of this exercise, we're determining how much energy the photon's light delivers to a single rod cell in the retina. The formula for intensity (I) is given as \( I = \frac{P}{A} \), where \( P \) is the power (or energy per time) absorbed, and \( A \) is the area over which the power is spread.

• First, we determined the energy absorbed by the rod which is given as \( 3.94 \times 10^{-19} \text{ J} \).
• The rod absorbs this energy when a photon hits it, so we assume the energy absorbed per unit time (power) is equal to this energy.
• The area of the rod's cross-section, calculated as \( \pi \times (1.0 \times 10^{-6})^2 \), is essential to find the intensity of light.

By using these values, the intensity becomes \( I \approx 1.25 \times 10^{-7} \text{ W/m}^2 \). This shows how concentrated the power is on the tiny area of the rod.

Photon Energy

Photon energy refers to the amount of energy carried by a single photon, which is a fundamental particle of light. The energy of a photon can be calculated using the formula \( E = h \cdot f \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), and \( f \) is the frequency of the photon.
For photons detected by the retina, the energy is specifically adjusted to the light's wavelength. In our exercise, we have used the energy directly provided as \( 3.94 \times 10^{-19} \text{ J} \) without calculating it from wavelength, but the concept remains core.

• Photon energy is key to understanding the amount of energy transferred in the form of light.
• This energy is what tiny receptor cells, like the rods in the retina, are sensitive to.
• The ability of rods to detect such low energy levels demonstrates the sensitivity of human vision in low light.

This photon energy is precisely what allows us to detect even minimal light levels under dark conditions.

Retina Rods

Retina rods are specialized photoreceptor cells located in the human eye. They play a pivotal role in our ability to see under low-light conditions, a phenomenon often referred to as scotopic vision. These cells are incredibly sensitive to light, which is why they can detect single photons.

• Rods contain a pigment called rhodopsin, which is extremely sensitive to light.
• Under dark conditions, rods can become even more sensitive, adjusting their capabilities to make the most of limited light.
• The diameter of a rod, approximately 0.0020 mm, highlights how small these cells are, yet they perform a critical function in vision.

Understanding how rods work helps explain why certain animal species, such as nocturnal animals, have heightened night vision compared to humans.

SI Unit Conversion

SI Unit Conversion is essential in scientific calculations as it standardizes the measurements in a way that is universally understandable. In our exercise, converting units was crucial to solving the problem accurately.

• The rods' diameter needed conversion from millimeters to meters. Given as 0.0020 mm, it was converted to meters by dividing by 1000, resulting in \( 2.0 \times 10^{-6} \text{ m} \).
• Using SI units for all measurements ensures consistency and precision in calculations.
• The conversion rules are simple: Length (meters), Mass (kilograms), Time (seconds), and other quantities are given their respective units.

By maintaining measurements in SI, we allow for easier collaboration, communication, and validation of scientific data worldwide.